1Thomas Liu, BE280A, UCSD, Fall 2005Bioengineering 280APrinciples of Biomedical ImagingFall Quarter 2005Noise and EstimationThomas Liu, BE280A, UCSD, Fall 2005What is Noise?Fluctuations in either the imaging system or the objectbeing imaged.Quantization Noise: Due to conversion from analogwaveform to digital number.Quantum Noise: Random fluctuation in the number ofphotons emittted and recorded.Thermal Noise: Random fluctuations present in allelectronic systems. Also, sample noise in MRIOther types: flicker, burst, avalanche - observed insemiconductor devices.Structured Noise: physiological sources, interferenceThomas Liu, BE280A, UCSD, Fall 2005Quantization NoiseSignal s(t)r(t) = s(t) + q(t)Although the noise is deterministic, it is useful to model the noise as a random process. Quantization noiseq(t)Quantized Signal r(t)2Thomas Liu, BE280A, UCSD, Fall 2005Physiological NoiseSignal Intensity Pulse Ox (finger)Cardiac component estimated in brain Image Number (4 Hz)Perfusion time series: Before Correction ( cc = 0.15)Perfusion time series: After Correction ( cc = 0.71)Perfusion Time SeriesThomas Liu, BE280A, UCSD, Fall 2005Noise and Image QualityPrince and Links 2005Thomas Liu, BE280A, UCSD, Fall 2005Thermal NoiseFluctuations in voltage across a resistor due to randomthermal motion of electrons.Described by J.B. Johnson in 1927 (therefore sometimescalled Johnson noise). Explained by H. Nyquist in 1928.€ V2= 4kT ⋅ R ⋅ BWVariance in VoltageResistanceBandwidthTemperature3Thomas Liu, BE280A, UCSD, Fall 2005Thermal Noise€ V2= 4kT ⋅ R⋅ BWAt room temperature, noise in a 1 kΩ resistor isV2/ BW =16 ×10−18 V2/ HzIn root mean squared form, this corresponds to V/BW = 4 nV/ Hz .Example : For BW = 250 kHz and 2 kΩ resistor, total noise voltage is 2 ⋅16 ×10-18⋅ 250 ×103= 4 µVThomas Liu, BE280A, UCSD, Fall 2005Thermal Noise€ Noise spectral density is independent of frequency upto 1013 Hz. Therefore it is a source of white noise.Amplitude distribution of the noise is Gaussian.Thomas Liu, BE280A, UCSD, Fall 2005Signal in MRI€ Recall the signal equation has the formsr(t) = M(x,y,z)e− t /T2( r )e− jω0texp − jγGτ( )0t∫⋅ r(τ)dτ ∫∫∫dxdydz€ Faraday's LawEMF = −∂φ∂tφ= Magnetic Flux = B1(x,y,z) ⋅ M( x, y,z)dV∫B04Thomas Liu, BE280A, UCSD, Fall 2005Signal in MRI€ Signal in the receiver coilsr(t) = jω0B1xyM( x, y,z)e− t /T2( r)e− jω0texp − jγGτ( )0t∫⋅ r(τ)dτ ∫dVRecall, total magnetization is proportional to B0Also ω0=γB0.Therefore, total signal is proportional to B02Thomas Liu, BE280A, UCSD, Fall 2005Noise in MRI€ Primary sources of noise are :1) Thermal noise of the receiver coil2) Thermal noise of the sample. Coil Resistance : At higher frequencies, the EM wavestend to travel along the surface of the conductor (skineffect). As a result, Rcoil ∝ ω01/2 ⇒ Ncoil2 ∝ ω01/2∝ B01/ 2Sample Noise : Noise is white, but differentiationprocess due to Faraday's law introduces a multiplicationby ω0. As a result, the noise variance from the sampleis proportional to ω02.Nsample2 ∝ω02∝ B02Thomas Liu, BE280A, UCSD, Fall 2005SNR in MRI€ SNR =signal amplitudestandard deviation of noise∝B02αB01/ 2+βB02If coil noise dominatesSNR ∝ B07 / 4If sample noise dominatesSNR ∝ B05Thomas Liu, BE280A, UCSD, Fall 2005Random Variables€ A random variable X is characterized by its cumulativedistribution function (CDF)Pr( X ≤ x) = FX(x)The derivative of the CDF is the probability densityfunction(pdf)fX(x) = dFX(x) / dxThe probability that X will take on values between twolimits x1 and x2 is Pr(x1≤ X ≤ x2) = FX(x2) − FX(x1) = fX(x)dxx1x2∫Thomas Liu, BE280A, UCSD, Fall 2005Mean and Variance€ µX= E[X]= xfX(x )dx−∞∞∫σX2= Var[X ]= E[ X −µX( )2]= (x −µX)2fX(x )dx−∞∞∫= E[X2] −µX2Thomas Liu, BE280A, UCSD, Fall 2005Gaussian Random Variable€ fX(x ) =12πσ2exp −(x −µ)2/ 2σ2( )( )µX=µσX2=σ26Thomas Liu, BE280A, UCSD, Fall 2005Independent Random Variables€ fX1,X2(x1, x2) = fX1(x1) fX2(x2)E[X1X2] = E[X1]E[ X2]Let Y = X1+ X2 then µY= E[Y]= E[ X1] + E[ X2]=µ1+µ2E[Y2] = E[X12] + 2E[X1]E[ X2] + E[ X22] = E[X12] + 2µ1µ2+ E[ X22]σY2= E[Y2] −µY2= E[ X12] + 2µ1µ2+ E[ X22] −µ12−µ22− 2µ1µ2=σX12+σX22Thomas Liu, BE280A, UCSD, Fall 2005Signal Averaging€ € We can improve SNR by averaging. Let y1= y0+ n1y2= y0+ n2The sum of the two measurements is 2y0+ n1+ n2( ).If the noise in the measurements is independent, then the variances sum and the total variance is 2σn2SNRTot=2y02σn= 2SNRoriginalIn general, SNR ∝ Nave∝ TimeThomas Liu, BE280A, UCSD, Fall 2005Random Processes € A random process is an indexed family of randomvariablesExamples : discrete : X1, X2,K, XNcontinuous : X(t)If all the random variables share the same pdf and take on values independently, the process is said tobe independent and identically distributed (iid).Example : unbiased coin tossesIf the joint statistics of the process do not varywith index, the process is said to be stationary.7Thomas Liu, BE280A, UCSD, Fall 2005Correlation and Covariance€ CorrelationR(t1,t2) = E( X(t1) X∗(t2))R(i, j) = E( XiXj)CovarianceC(t1,t2) = E X(t1) − X (t1)( )X(t2) − X (t2)( )∗ C(i, j) = E Xi− X i( )Xj− X j( )( )Thomas Liu, BE280A, UCSD, Fall 2005Stationary Process€ For a wide - sense stationary processE(X(t)) =µR(t,t2) = R(τ) = E(X(t)X∗(t +τ)) for τ= t2− tR(i, j) = R(m) = E (XiXi+m) for m = j − iExample : White noise processE[X] = 0C (τ) =σ2δ(τ)Thomas Liu, BE280A, UCSD, Fall 2005Power Spectral Density€ For a wide - sense stationary process, we can definethe Power Spectral Density as : SX( f ) = F R(τ){ } for a continuous random process or SX( f ) = F R(m){ } for a discrete random process Example : White noiseSX( f ) = Fσ2δ(τ){ }=σ28Thomas Liu, BE280A, UCSD, Fall 2005Vector Notation € X =X1X2MXN R = E(XXH) = EX1X1∗X1X2∗L X1XN∗X2X1∗X2X2∗L X2XN∗M M O MXNX1∗XNX2∗L XNXN∗ Thomas Liu, BE280A, UCSD, Fall 2005Example € X denotes a stationary random process with mean zeroand correlation R[m] =σ2δ[m]R = E(XXH) =σ20 L 00σ2L 0M M O M0 0 Lσ2 =σ2IThomas Liu, BE280A, UCSD, Fall
View Full Document
Unlocking...